$T\bar{T}$ deformed partition functions

TL;DR

The paper demonstrates that $T\bar{T}$-deformed CFTs retain non-holomorphic modular structure in their torus partition functions, with the full partition function obeying $Z(\tau,\bar{\tau}|\lambda)=Z\left( \frac{a\tau+b}{c\tau+d}, \frac{a\bar{\tau}+b}{c\bar{\tau}+d} \big| \frac{\lambda}{|c\tau+d|} \right)$. Using the deformed energy spectrum and a diffusion equation, the authors derive a perturbative, modular-covariant expansion in $\lambda$, show that each perturbative piece $Z_p$ is a non-holomorphic modular form of weight $2p$, and prove the modular property by a recursion. They verify the framework explicitly for the deformed free boson, where perturbative corrections are expressible in terms of Eisenstein series, and they extract the high-energy density of states, which interpolates between Cardy-like and Hagedorn-like growth. The work also identifies a BPS-like protected sector with invariant elliptic genus and discusses holographic interpretations via a finite-cutoff AdS$_3$ dual, including implications for the deformed vacuum character and one-loop determinants.

Abstract

We demonstrate the presence of modular properties in partition functions of $T\bar{T}$ deformed conformal field theories. These properties are verified explicitly for the deformed free boson. The modular features facilitate a derivation of the asymptotic density of states in these theories, which turns out to interpolate between Cardy and Hagedorn behaviours. We also point out a sub-sector of the spectrum that remains undeformed under the $T\bar{T}$ flow. Finally, we comment on the deformation of the CFT vacuum character and its implications for the holographic dual.